Time-Series – Understanding Unit Root Test and Cointegration

Question

maybe someone can help me with my data.

I analyse how macroeconomic indicators affect stock index. For this analysis I prefer VAR model.In my case data of all variables are non-stationary – I have checked it by plots and also with adf test.Also I made unit root test and it says that all variables have unit root. After that I decided to apply differencing method with first order. After that with adf test I got results that all variables are stationary, because p-value is less than 0.05 but unit root test shows that some variables still have unit root.

My question is should I also apply cointegration test in this case after I found that differenced data still has unit roots? Or should it be applied before making changes as differencing in data? Basicly through a lot of sources I don't understand when should I use unit root test and cointegration test when I want to create VAR model.

Thanks in advance!

Answer 1

Let us say the highest order of integration is $d$; usually, $d=1$ though sometimes $d=2$.

1. If $d=1$ and there is a single series that is I(1) while the other ones are I(0), you take first differences of that variable and model that together with the other variables using a VAR.
2. If $d=1$ and there are two or more I(1) variables, you test for cointegration between them. If they are not cointegrated, you take first differences of them and model them together with I(0) variables using a VAR. If, on the other hand, the I(1) variables are cointegrated, you model them using VECM; you also include the I(0) variables on the side as in this answer. (Watch out that the left hand side of your equations is of the same order of integration as the right hand side.)
3. If $d=2$ and there is a single series that is I(2) while the other ones are I(1) and I(0), you take first differences of the I(2) variable and then proceed as in 1., now treating the differenced I(2) variable as a primitive.
4. If $d=2$ and there are two or more I(2) variables, you test for cointegration between them. If they are not cointegrated, you take first differences of them and model them together with the other variables as in 1. If, on the other hand, the I(2) variables are cointegrated, you obtain the error correction term(s) and the first-differences of all the I(2) variables and proceed as in (2). This is a bit terse, but trying to cover all cases in detail is quite tedious. (Again, watch out that the left hand side of your equations is of the same order of integration as the right hand side.)